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❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❏✉♥❡ ✶✻✱ ✷✵✷✵

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✶ ✴ ✷✽

❈r②♣t♦❧♦❣②

❙❝✐❡♥❝❡ ❝♦♥❝❡r♥❡❞ ✇✐t❤ ❞❛t❛ ❝♦♠♠✉♥✐❝❛t✐♦♥ ❛♥❞ st♦r❛❣❡ ✐♥ s❡❝✉r❡ ❛♥❞ ✉s✉❛❧❧② s❡❝r❡t ❢♦r♠✳ ❈r②♣t♦❣r❛♣❤② ✖ s❡❝r❡t ✇r✐t✐♥❣ ✭♦r ♦t❤❡r ♠❡t❤♦❞s ♦❢ ❤✐❞✐♥❣ ✐♥❢♦r♠❛t✐♦♥✮ ❈r②♣t❛♥❛❧②s✐s ✖ r❡❛❞✐♥❣ s❡❝r❡t ♠❡ss❛❣❡s ✭✇✐t❤♦✉t ❦♥♦✇❧❡❞❣❡ ♦❢ ❡♥❝r②♣t✐♦♥ ❦❡②✮✳

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✷ ✴ ✷✽

P❛st → ❚♦❞❛② → ❋✉t✉r❡

❈❧❛ss✐❝ ❝r②♣t♦❣r❛♣❤② ✖ ♠❛♥✐♣✉❧❛t✐♦♥ ✭s✉❜st✐t✉t✐♥❣ ❛♥❞ ♣❡r♠✉t✐♥❣✮ ♦❢ s②♠❜♦❧s ✭❧❡tt❡rs✮✱ ❝♦❞❡❜♦♦❦s✳ ▼♦❞❡r♥ ❝r②♣t♦❣r❛♣❤② ✖ ❝♦♠♣✉t❡rs ❛♥❞ ♠❛t❤❡♠❛t✐❝s✳ ◗✉❛♥t✉♠ ❝r②♣t♦❣r❛♣❤② ✖ q✉❛♥t✉♠ ♣❤②s✐❝s ❛♥❞ q✉❛♥t✉♠ ❝♦♠♣✉t❡rs✳

q✉❛♥t✉♠ ❝r②♣t❛♥❛❧②s✐s ✭❙❤♦r✬s ❛❧❣♦r✐t❤♠✮✱ ❦❡② ❡①❝❤❛♥❣❡ ♠❡t❤♦❞s✱ ❞❡t❡❝t✐♦♥ ♦❢ s❡❝✉r✐t② ❜r❡❡❝❤❡s✳

P♦stq✉❛♥t✉♠ ❝r②♣t♦❣r❛♣❤② ✖ ❛❧❣♦r✐t❤♠s r❡s✐st❛♥t t♦ q✉❛♥t✉♠ ❝r②♣t❛♥❛❧②s✐s

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✸ ✴ ✷✽

P✉❜❧✐❝✲❦❡② ❝r②♣t♦❣r❛♣❤②

❙②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤②✿ ❊♥❝r②♣t✐♦♥ ❛♥❞ ❞❡❝r②♣t✐♦♥ ✉s❡s t❤❡ s❛♠❡ ❦❡②✳ P✉❜❧✐❝✲❦❡② ✭❛s②♠♠❡tr✐❝✮ ❝r②♣t♦❣r❛♣❤②✿ ❑❡② ❢♦r ❡♥❝r②♣t✐♦♥ ✭♣✉❜❧✐❝✮ ✐s ❞✐✛❡r❡♥t t❤❛♥ ❦❡② ❢♦r ❞❡❝r②♣t✐♦♥ ✭♣r✐✈❛t❡✮✳ ❊♥❝r②♣t✐♦♥✴❞❡❝r②♣t✐♦♥ ❛♥❞ ❞✐❣✐t❛❧ s✐❣♥❛t✉r❡✳ P✉❜❧✐❝ ❦❡② ✐s ✉s❡❞ t♦ ❡♥❝r②♣t ❛♥❞ ✈❡r✐❢② s✐❣♥❛t✉r❡✳ Pr✐✈❛t❡ ❦❡② ✐s ✉s❡❞ t♦ ❞❡❝r②♣t ❛♥❞ t♦ ❝r❡❛t❡ ❞✐❣✐t❛❧ s✐❣♥❛t✉r❡✳

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✹ ✴ ✷✽

P✉❜❧✐❝✲❦❡② ❝r②♣t♦❣r❛♣❤②

❙②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤②✿ ❊♥❝r②♣t✐♦♥ ❛♥❞ ❞❡❝r②♣t✐♦♥ ✉s❡s t❤❡ s❛♠❡ ❦❡②✳ P✉❜❧✐❝✲❦❡② ✭❛s②♠♠❡tr✐❝✮ ❝r②♣t♦❣r❛♣❤②✿ ❑❡② ❢♦r ❡♥❝r②♣t✐♦♥ ✭♣✉❜❧✐❝✮ ✐s ❞✐✛❡r❡♥t t❤❛♥ ❦❡② ❢♦r ❞❡❝r②♣t✐♦♥ ✭♣r✐✈❛t❡✮✳ ❊♥❝r②♣t✐♦♥✴❞❡❝r②♣t✐♦♥ ❛♥❞ ❞✐❣✐t❛❧ s✐❣♥❛t✉r❡✳ P✉❜❧✐❝ ❦❡② ✐s ✉s❡❞ t♦ ❡♥❝r②♣t ❛♥❞ ✈❡r✐❢② s✐❣♥❛t✉r❡✳ Pr✐✈❛t❡ ❦❡② ✐s ✉s❡❞ t♦ ❞❡❝r②♣t ❛♥❞ t♦ ❝r❡❛t❡ ❞✐❣✐t❛❧ s✐❣♥❛t✉r❡✳

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✹ ✴ ✷✽

P✉❜❧✐❝✲❦❡② ❝r②♣t♦❣r❛♣❤②

❙②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤②✿ ❊♥❝r②♣t✐♦♥ ❛♥❞ ❞❡❝r②♣t✐♦♥ ✉s❡s t❤❡ s❛♠❡ ❦❡②✳ P✉❜❧✐❝✲❦❡② ✭❛s②♠♠❡tr✐❝✮ ❝r②♣t♦❣r❛♣❤②✿ ❑❡② ❢♦r ❡♥❝r②♣t✐♦♥ ✭♣✉❜❧✐❝✮ ✐s ❞✐✛❡r❡♥t t❤❛♥ ❦❡② ❢♦r ❞❡❝r②♣t✐♦♥ ✭♣r✐✈❛t❡✮✳ ❊♥❝r②♣t✐♦♥✴❞❡❝r②♣t✐♦♥ ❛♥❞ ❞✐❣✐t❛❧ s✐❣♥❛t✉r❡✳ P✉❜❧✐❝ ❦❡② ✐s ✉s❡❞ t♦ ❡♥❝r②♣t ❛♥❞ ✈❡r✐❢② s✐❣♥❛t✉r❡✳ Pr✐✈❛t❡ ❦❡② ✐s ✉s❡❞ t♦ ❞❡❝r②♣t ❛♥❞ t♦ ❝r❡❛t❡ ❞✐❣✐t❛❧ s✐❣♥❛t✉r❡✳

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✹ ✴ ✷✽

P✉❜❧✐❝✲❦❡② ❝r②♣t♦❣r❛♣❤②

❙②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤②✿ ❊♥❝r②♣t✐♦♥ ❛♥❞ ❞❡❝r②♣t✐♦♥ ✉s❡s t❤❡ s❛♠❡ ❦❡②✳ P✉❜❧✐❝✲❦❡② ✭❛s②♠♠❡tr✐❝✮ ❝r②♣t♦❣r❛♣❤②✿ ❑❡② ❢♦r ❡♥❝r②♣t✐♦♥ ✭♣✉❜❧✐❝✮ ✐s ❞✐✛❡r❡♥t t❤❛♥ ❦❡② ❢♦r ❞❡❝r②♣t✐♦♥ ✭♣r✐✈❛t❡✮✳ ❊♥❝r②♣t✐♦♥✴❞❡❝r②♣t✐♦♥ ❛♥❞ ❞✐❣✐t❛❧ s✐❣♥❛t✉r❡✳ P✉❜❧✐❝ ❦❡② ✐s ✉s❡❞ t♦ ❡♥❝r②♣t ❛♥❞ ✈❡r✐❢② s✐❣♥❛t✉r❡✳ Pr✐✈❛t❡ ❦❡② ✐s ✉s❡❞ t♦ ❞❡❝r②♣t ❛♥❞ t♦ ❝r❡❛t❡ ❞✐❣✐t❛❧ s✐❣♥❛t✉r❡✳

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✹ ✴ ✷✽

P✉❜❧✐❝✲❦❡② ❝r②♣t♦❣r❛♣❤②

❙②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤②✿ ❊♥❝r②♣t✐♦♥ ❛♥❞ ❞❡❝r②♣t✐♦♥ ✉s❡s t❤❡ s❛♠❡ ❦❡②✳ P✉❜❧✐❝✲❦❡② ✭❛s②♠♠❡tr✐❝✮ ❝r②♣t♦❣r❛♣❤②✿ ❑❡② ❢♦r ❡♥❝r②♣t✐♦♥ ✭♣✉❜❧✐❝✮ ✐s ❞✐✛❡r❡♥t t❤❛♥ ❦❡② ❢♦r ❞❡❝r②♣t✐♦♥ ✭♣r✐✈❛t❡✮✳ ❊♥❝r②♣t✐♦♥✴❞❡❝r②♣t✐♦♥ ❛♥❞ ❞✐❣✐t❛❧ s✐❣♥❛t✉r❡✳ P✉❜❧✐❝ ❦❡② ✐s ✉s❡❞ t♦ ❡♥❝r②♣t ❛♥❞ ✈❡r✐❢② s✐❣♥❛t✉r❡✳ Pr✐✈❛t❡ ❦❡② ✐s ✉s❡❞ t♦ ❞❡❝r②♣t ❛♥❞ t♦ ❝r❡❛t❡ ❞✐❣✐t❛❧ s✐❣♥❛t✉r❡✳

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✹ ✴ ✷✽

P✉❜❧✐❝✲❦❡② ❝r②♣t♦❣r❛♣❤②

❙②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤②✿ ❊♥❝r②♣t✐♦♥ ❛♥❞ ❞❡❝r②♣t✐♦♥ ✉s❡s t❤❡ s❛♠❡ ❦❡②✳ P✉❜❧✐❝✲❦❡② ✭❛s②♠♠❡tr✐❝✮ ❝r②♣t♦❣r❛♣❤②✿ ❑❡② ❢♦r ❡♥❝r②♣t✐♦♥ ✭♣✉❜❧✐❝✮ ✐s ❞✐✛❡r❡♥t t❤❛♥ ❦❡② ❢♦r ❞❡❝r②♣t✐♦♥ ✭♣r✐✈❛t❡✮✳ ❊♥❝r②♣t✐♦♥✴❞❡❝r②♣t✐♦♥ ❛♥❞ ❞✐❣✐t❛❧ s✐❣♥❛t✉r❡✳ P✉❜❧✐❝ ❦❡② ✐s ✉s❡❞ t♦ ❡♥❝r②♣t ❛♥❞ ✈❡r✐❢② s✐❣♥❛t✉r❡✳ Pr✐✈❛t❡ ❦❡② ✐s ✉s❡❞ t♦ ❞❡❝r②♣t ❛♥❞ t♦ ❝r❡❛t❡ ❞✐❣✐t❛❧ s✐❣♥❛t✉r❡✳

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✹ ✴ ✷✽

P✉❜❧✐❝✲❦❡② ❝r②♣t♦❣r❛♣❤②

❙②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤②✿ ❊♥❝r②♣t✐♦♥ ❛♥❞ ❞❡❝r②♣t✐♦♥ ✉s❡s t❤❡ s❛♠❡ ❦❡②✳ P✉❜❧✐❝✲❦❡② ✭❛s②♠♠❡tr✐❝✮ ❝r②♣t♦❣r❛♣❤②✿ ❑❡② ❢♦r ❡♥❝r②♣t✐♦♥ ✭♣✉❜❧✐❝✮ ✐s ❞✐✛❡r❡♥t t❤❛♥ ❦❡② ❢♦r ❞❡❝r②♣t✐♦♥ ✭♣r✐✈❛t❡✮✳ ❊♥❝r②♣t✐♦♥✴❞❡❝r②♣t✐♦♥ ❛♥❞ ❞✐❣✐t❛❧ s✐❣♥❛t✉r❡✳ P✉❜❧✐❝ ❦❡② ✐s ✉s❡❞ t♦ ❡♥❝r②♣t ❛♥❞ ✈❡r✐❢② s✐❣♥❛t✉r❡✳ Pr✐✈❛t❡ ❦❡② ✐s ✉s❡❞ t♦ ❞❡❝r②♣t ❛♥❞ t♦ ❝r❡❛t❡ ❞✐❣✐t❛❧ s✐❣♥❛t✉r❡✳

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✹ ✴ ✷✽

❍✐st♦r② ♦❢ ♣✉❜❧✐❝✲❦❡② ❝r②♣t♦❣r❛♣❤②

✶ ❉✐✣❡✲❍❡❧❧♠❛♥ ❦❡② ❡①❝❤❛♥❣❡ ♣r♦t♦❝♦❧✿ ✶✾✼✻ ✷ ❘❙❆ ❛❧❣♦r✐t❤♠✿ ✶✾✼✼ ✸ ❊❧●❛♠❛❧ ❝r②♣t♦s②st❡♠✿ ✶✾✽✺ ✹ ❇r✐t✐s❤ ■♥t❡❧❧✐❣❡♥❝❡ ✭●❈❍◗✮✿ ✶✾✼✵ ✕ ✶✾✼✸ ✭❝❧❛ss✐✜❡❞ ❢♦r ✷✺ ②❡❛rs✮ P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✺ ✴ ✷✽

❉✐✣❡✲❍❡❧❧♠❛♥✲▼❡r❦❧❡ ♣r♦t♦❝♦❧

❑❡② ❡①❝❤❛♥❣❡ ❉✐✣❡✲❍❡❧❧♠❛♥✲▼❡r❦❧❡ ♣r♦t♦❝♦❧

❋✐rst ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ❛❧❣♦r✐t❤♠ ✭✶✾✼✻✮✳ ■♥✈❡♥t❡❞ ❜② ▼❛rt✐♥ ❍❡❧❧♠❛♥ ❛♥❞ ❲❤✐t✜❡❧❞ ❉✐✣✱ ❛♥❞ ✐♥❞❡♣❡♥❞❡♥t❧② ❜② ❘❛❧♣❤ ▼❡r❦❧❡✳ ❖♥❧② ❦❡② ❡①❝❤❛♥❣❡✳ ❙❡❝✉r✐t② ❜❛s❡❞ ♦♥ t❤❡ ❞✐✣❝✉❧t② ♦❢ ❝❛❧❝✉❧❛t✐♥❣ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠s ✭❉✐s❝r❡t❡ ▲♦❣❛r✐t❤♠ Pr♦❜❧❡♠✮✳

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✻ ✴ ✷✽

❉✐✣❡✲❍❡❧❧♠❛♥✲▼❡r❦❧❡ ♣r♦t♦❝♦❧

❍♦✇ ✐t ✇♦r❦s❄

✶ ❆❧✐❝❡ ❛♥❞ ❇♦❜ s❡❧❡❝t ❧❛r❣❡ ♣r✐♠❡ ♥✉♠❜❡r n ❛♥❞ ❛ ♥✉♠❜❡r g s✉❝❤

t❤❛t 1 < g < n✳

✷ ❆❧✐❝❡ ❝❤♦♦s❡s ❧❛r❣❡ ♥✉♠❜❡r

❛♥❞ s❡♥❞ t♦ ❇♦❜ r❡s✉❧t ♦❢ ❝❛❧❝✉❧❛t✐♦♥ ✳

✸ ❇♦❜ ❛❧s♦ ❝❤♦♦s❡s ❧❛r❣❡ ♥✉♠❜❡r

❛♥❞ s❡♥❞ ❆❧✐❝❡ r❡s✉❧t ♦❢ ❝❛❧❝✉❧❛t✐♦♥ ✳

✹ ❆❧✐❝❡ ❝♦♠♣✉t❡s

✳

✺ ❇♦❜ ❝♦♠♣✉t❡s

✳

✻ ❆❧✐❝❡ ❛♥❞ ❇♦❜ ♥♦✇ s❤❛r❡ ❛ s❡❝r❡t ✭♥✉♠❜❡r✮

✳ ❊✈❡ ✭❡❛✈❡s❞r♦♣♣❡r✮ ❦♥♦✇s ✱ ✱ ✱ ■♥ ♦r❞❡r t♦ ❦♥♦✇ ❛♥❞ s❤❡ ♠✉st ❝❛❧❝✉❧❛t❡ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠ ✳

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✼ ✴ ✷✽

❉✐✣❡✲❍❡❧❧♠❛♥✲▼❡r❦❧❡ ♣r♦t♦❝♦❧

❍♦✇ ✐t ✇♦r❦s❄

✶ ❆❧✐❝❡ ❛♥❞ ❇♦❜ s❡❧❡❝t ❧❛r❣❡ ♣r✐♠❡ ♥✉♠❜❡r n ❛♥❞ ❛ ♥✉♠❜❡r g s✉❝❤

t❤❛t 1 < g < n✳

✷ ❆❧✐❝❡ ❝❤♦♦s❡s ❧❛r❣❡ ♥✉♠❜❡r a ❛♥❞ s❡♥❞ t♦ ❇♦❜ r❡s✉❧t ♦❢ ❝❛❧❝✉❧❛t✐♦♥

x = gamod n✳

✸ ❇♦❜ ❛❧s♦ ❝❤♦♦s❡s ❧❛r❣❡ ♥✉♠❜❡r

❛♥❞ s❡♥❞ ❆❧✐❝❡ r❡s✉❧t ♦❢ ❝❛❧❝✉❧❛t✐♦♥ ✳

✹ ❆❧✐❝❡ ❝♦♠♣✉t❡s

✳

✺ ❇♦❜ ❝♦♠♣✉t❡s

✳

✻ ❆❧✐❝❡ ❛♥❞ ❇♦❜ ♥♦✇ s❤❛r❡ ❛ s❡❝r❡t ✭♥✉♠❜❡r✮

✳ ❊✈❡ ✭❡❛✈❡s❞r♦♣♣❡r✮ ❦♥♦✇s ✱ ✱ ✱ ■♥ ♦r❞❡r t♦ ❦♥♦✇ ❛♥❞ s❤❡ ♠✉st ❝❛❧❝✉❧❛t❡ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠ ✳

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✼ ✴ ✷✽

❉✐✣❡✲❍❡❧❧♠❛♥✲▼❡r❦❧❡ ♣r♦t♦❝♦❧

❍♦✇ ✐t ✇♦r❦s❄

✶ ❆❧✐❝❡ ❛♥❞ ❇♦❜ s❡❧❡❝t ❧❛r❣❡ ♣r✐♠❡ ♥✉♠❜❡r n ❛♥❞ ❛ ♥✉♠❜❡r g s✉❝❤

t❤❛t 1 < g < n✳

✷ ❆❧✐❝❡ ❝❤♦♦s❡s ❧❛r❣❡ ♥✉♠❜❡r a ❛♥❞ s❡♥❞ t♦ ❇♦❜ r❡s✉❧t ♦❢ ❝❛❧❝✉❧❛t✐♦♥

x = gamod n✳

✸ ❇♦❜ ❛❧s♦ ❝❤♦♦s❡s ❧❛r❣❡ ♥✉♠❜❡r b ❛♥❞ s❡♥❞ ❆❧✐❝❡ r❡s✉❧t ♦❢

❝❛❧❝✉❧❛t✐♦♥ y = gbmod n✳

✹ ❆❧✐❝❡ ❝♦♠♣✉t❡s

✳

✺ ❇♦❜ ❝♦♠♣✉t❡s

✳

✻ ❆❧✐❝❡ ❛♥❞ ❇♦❜ ♥♦✇ s❤❛r❡ ❛ s❡❝r❡t ✭♥✉♠❜❡r✮

✳ ❊✈❡ ✭❡❛✈❡s❞r♦♣♣❡r✮ ❦♥♦✇s ✱ ✱ ✱ ■♥ ♦r❞❡r t♦ ❦♥♦✇ ❛♥❞ s❤❡ ♠✉st ❝❛❧❝✉❧❛t❡ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠ ✳

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✼ ✴ ✷✽

❉✐✣❡✲❍❡❧❧♠❛♥✲▼❡r❦❧❡ ♣r♦t♦❝♦❧

❍♦✇ ✐t ✇♦r❦s❄

✶ ❆❧✐❝❡ ❛♥❞ ❇♦❜ s❡❧❡❝t ❧❛r❣❡ ♣r✐♠❡ ♥✉♠❜❡r n ❛♥❞ ❛ ♥✉♠❜❡r g s✉❝❤

t❤❛t 1 < g < n✳

✷ ❆❧✐❝❡ ❝❤♦♦s❡s ❧❛r❣❡ ♥✉♠❜❡r a ❛♥❞ s❡♥❞ t♦ ❇♦❜ r❡s✉❧t ♦❢ ❝❛❧❝✉❧❛t✐♦♥

x = gamod n✳

✸ ❇♦❜ ❛❧s♦ ❝❤♦♦s❡s ❧❛r❣❡ ♥✉♠❜❡r b ❛♥❞ s❡♥❞ ❆❧✐❝❡ r❡s✉❧t ♦❢

❝❛❧❝✉❧❛t✐♦♥ y = gbmod n✳

✹ ❆❧✐❝❡ ❝♦♠♣✉t❡s kA = yamod n✳ ✺ ❇♦❜ ❝♦♠♣✉t❡s

✳

✻ ❆❧✐❝❡ ❛♥❞ ❇♦❜ ♥♦✇ s❤❛r❡ ❛ s❡❝r❡t ✭♥✉♠❜❡r✮

✳ ❊✈❡ ✭❡❛✈❡s❞r♦♣♣❡r✮ ❦♥♦✇s ✱ ✱ ✱ ■♥ ♦r❞❡r t♦ ❦♥♦✇ ❛♥❞ s❤❡ ♠✉st ❝❛❧❝✉❧❛t❡ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠ ✳

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✼ ✴ ✷✽

❉✐✣❡✲❍❡❧❧♠❛♥✲▼❡r❦❧❡ ♣r♦t♦❝♦❧

❍♦✇ ✐t ✇♦r❦s❄

✶ ❆❧✐❝❡ ❛♥❞ ❇♦❜ s❡❧❡❝t ❧❛r❣❡ ♣r✐♠❡ ♥✉♠❜❡r n ❛♥❞ ❛ ♥✉♠❜❡r g s✉❝❤

t❤❛t 1 < g < n✳

✷ ❆❧✐❝❡ ❝❤♦♦s❡s ❧❛r❣❡ ♥✉♠❜❡r a ❛♥❞ s❡♥❞ t♦ ❇♦❜ r❡s✉❧t ♦❢ ❝❛❧❝✉❧❛t✐♦♥

x = gamod n✳

✸ ❇♦❜ ❛❧s♦ ❝❤♦♦s❡s ❧❛r❣❡ ♥✉♠❜❡r b ❛♥❞ s❡♥❞ ❆❧✐❝❡ r❡s✉❧t ♦❢

❝❛❧❝✉❧❛t✐♦♥ y = gbmod n✳

✹ ❆❧✐❝❡ ❝♦♠♣✉t❡s kA = yamod n✳ ✺ ❇♦❜ ❝♦♠♣✉t❡s kB = xbmod n✳ ✻ ❆❧✐❝❡ ❛♥❞ ❇♦❜ ♥♦✇ s❤❛r❡ ❛ s❡❝r❡t ✭♥✉♠❜❡r✮

✳ ❊✈❡ ✭❡❛✈❡s❞r♦♣♣❡r✮ ❦♥♦✇s ✱ ✱ ✱ ■♥ ♦r❞❡r t♦ ❦♥♦✇ ❛♥❞ s❤❡ ♠✉st ❝❛❧❝✉❧❛t❡ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠ ✳

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✼ ✴ ✷✽

❉✐✣❡✲❍❡❧❧♠❛♥✲▼❡r❦❧❡ ♣r♦t♦❝♦❧

❍♦✇ ✐t ✇♦r❦s❄

✶ ❆❧✐❝❡ ❛♥❞ ❇♦❜ s❡❧❡❝t ❧❛r❣❡ ♣r✐♠❡ ♥✉♠❜❡r n ❛♥❞ ❛ ♥✉♠❜❡r g s✉❝❤

t❤❛t 1 < g < n✳

✷ ❆❧✐❝❡ ❝❤♦♦s❡s ❧❛r❣❡ ♥✉♠❜❡r a ❛♥❞ s❡♥❞ t♦ ❇♦❜ r❡s✉❧t ♦❢ ❝❛❧❝✉❧❛t✐♦♥

x = gamod n✳

✸ ❇♦❜ ❛❧s♦ ❝❤♦♦s❡s ❧❛r❣❡ ♥✉♠❜❡r b ❛♥❞ s❡♥❞ ❆❧✐❝❡ r❡s✉❧t ♦❢

❝❛❧❝✉❧❛t✐♦♥ y = gbmod n✳

✹ ❆❧✐❝❡ ❝♦♠♣✉t❡s kA = yamod n✳ ✺ ❇♦❜ ❝♦♠♣✉t❡s kB = xbmod n✳ ✻ ❆❧✐❝❡ ❛♥❞ ❇♦❜ ♥♦✇ s❤❛r❡ ❛ s❡❝r❡t ✭♥✉♠❜❡r✮kA = kB = gabmod n✳

❊✈❡ ✭❡❛✈❡s❞r♦♣♣❡r✮ ❦♥♦✇s ✱ ✱ ✱ ■♥ ♦r❞❡r t♦ ❦♥♦✇ ❛♥❞ s❤❡ ♠✉st ❝❛❧❝✉❧❛t❡ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠ ✳

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✼ ✴ ✷✽

❉✐✣❡✲❍❡❧❧♠❛♥✲▼❡r❦❧❡ ♣r♦t♦❝♦❧

❍♦✇ ✐t ✇♦r❦s❄

✶ ❆❧✐❝❡ ❛♥❞ ❇♦❜ s❡❧❡❝t ❧❛r❣❡ ♣r✐♠❡ ♥✉♠❜❡r n ❛♥❞ ❛ ♥✉♠❜❡r g s✉❝❤

t❤❛t 1 < g < n✳

✷ ❆❧✐❝❡ ❝❤♦♦s❡s ❧❛r❣❡ ♥✉♠❜❡r a ❛♥❞ s❡♥❞ t♦ ❇♦❜ r❡s✉❧t ♦❢ ❝❛❧❝✉❧❛t✐♦♥

x = gamod n✳

✸ ❇♦❜ ❛❧s♦ ❝❤♦♦s❡s ❧❛r❣❡ ♥✉♠❜❡r b ❛♥❞ s❡♥❞ ❆❧✐❝❡ r❡s✉❧t ♦❢

❝❛❧❝✉❧❛t✐♦♥ y = gbmod n✳

✹ ❆❧✐❝❡ ❝♦♠♣✉t❡s kA = yamod n✳ ✺ ❇♦❜ ❝♦♠♣✉t❡s kB = xbmod n✳ ✻ ❆❧✐❝❡ ❛♥❞ ❇♦❜ ♥♦✇ s❤❛r❡ ❛ s❡❝r❡t ✭♥✉♠❜❡r✮kA = kB = gabmod n✳

❊✈❡ ✭❡❛✈❡s❞r♦♣♣❡r✮ ❦♥♦✇s x✱ y✱ g✱ n ■♥ ♦r❞❡r t♦ ❦♥♦✇ ❛♥❞ s❤❡ ♠✉st ❝❛❧❝✉❧❛t❡ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠ ✳

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✼ ✴ ✷✽

❉✐✣❡✲❍❡❧❧♠❛♥✲▼❡r❦❧❡ ♣r♦t♦❝♦❧

❍♦✇ ✐t ✇♦r❦s❄

✶ ❆❧✐❝❡ ❛♥❞ ❇♦❜ s❡❧❡❝t ❧❛r❣❡ ♣r✐♠❡ ♥✉♠❜❡r n ❛♥❞ ❛ ♥✉♠❜❡r g s✉❝❤

t❤❛t 1 < g < n✳

✷ ❆❧✐❝❡ ❝❤♦♦s❡s ❧❛r❣❡ ♥✉♠❜❡r a ❛♥❞ s❡♥❞ t♦ ❇♦❜ r❡s✉❧t ♦❢ ❝❛❧❝✉❧❛t✐♦♥

x = gamod n✳

✸ ❇♦❜ ❛❧s♦ ❝❤♦♦s❡s ❧❛r❣❡ ♥✉♠❜❡r b ❛♥❞ s❡♥❞ ❆❧✐❝❡ r❡s✉❧t ♦❢

❝❛❧❝✉❧❛t✐♦♥ y = gbmod n✳

✹ ❆❧✐❝❡ ❝♦♠♣✉t❡s kA = yamod n✳ ✺ ❇♦❜ ❝♦♠♣✉t❡s kB = xbmod n✳ ✻ ❆❧✐❝❡ ❛♥❞ ❇♦❜ ♥♦✇ s❤❛r❡ ❛ s❡❝r❡t ✭♥✉♠❜❡r✮kA = kB = gabmod n✳

❊✈❡ ✭❡❛✈❡s❞r♦♣♣❡r✮ ❦♥♦✇s x✱ y✱ g✱ n ■♥ ♦r❞❡r t♦ ❦♥♦✇ ❛♥❞ s❤❡ ♠✉st ❝❛❧❝✉❧❛t❡ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠ ✳

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✼ ✴ ✷✽

❉✐✣❡✲❍❡❧❧♠❛♥✲▼❡r❦❧❡ ♣r♦t♦❝♦❧

❍♦✇ ✐t ✇♦r❦s❄

✶ ❆❧✐❝❡ ❛♥❞ ❇♦❜ s❡❧❡❝t ❧❛r❣❡ ♣r✐♠❡ ♥✉♠❜❡r n ❛♥❞ ❛ ♥✉♠❜❡r g s✉❝❤

t❤❛t 1 < g < n✳

✷ ❆❧✐❝❡ ❝❤♦♦s❡s ❧❛r❣❡ ♥✉♠❜❡r a ❛♥❞ s❡♥❞ t♦ ❇♦❜ r❡s✉❧t ♦❢ ❝❛❧❝✉❧❛t✐♦♥

x = gamod n✳

✸ ❇♦❜ ❛❧s♦ ❝❤♦♦s❡s ❧❛r❣❡ ♥✉♠❜❡r b ❛♥❞ s❡♥❞ ❆❧✐❝❡ r❡s✉❧t ♦❢

❝❛❧❝✉❧❛t✐♦♥ y = gbmod n✳

✹ ❆❧✐❝❡ ❝♦♠♣✉t❡s kA = yamod n✳ ✺ ❇♦❜ ❝♦♠♣✉t❡s kB = xbmod n✳ ✻ ❆❧✐❝❡ ❛♥❞ ❇♦❜ ♥♦✇ s❤❛r❡ ❛ s❡❝r❡t ✭♥✉♠❜❡r✮kA = kB = gabmod n✳

❊✈❡ ✭❡❛✈❡s❞r♦♣♣❡r✮ ❦♥♦✇s x✱ y✱ g✱ n → ■♥ ♦r❞❡r t♦ ❦♥♦✇ a ❛♥❞ b s❤❡ ♠✉st ❝❛❧❝✉❧❛t❡ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠✳

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✼ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❘❙❆ ❈r②♣t♦s②st❡♠

❘✐✈❡st✱ ❙❤❛♠✐r✱ ❆❞❧❡♠❛♥ ✭✶✾✼✼ r✳✮ ❙❡❝✉r✐t② ❜❛s❡❞ ♦♥ t❤❡ ❞✐✣❝✉❧t② ♦❢ ✐♥t❡❣❡r ❢❛❝t♦r✐③❛t✐♦♥✳

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✽ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❘❙❆ ❈r②♣t♦s②st❡♠

❘✐✈❡st✱ ❙❤❛♠✐r✱ ❆❞❧❡♠❛♥ ✭✷✵✵✸ r✳✮ ❙❡❝✉r✐t② ❜❛s❡❞ ♦♥ t❤❡ ❞✐✣❝✉❧t② ♦❢ ✐♥t❡❣❡r ❢❛❝t♦r✐③❛t✐♦♥✳

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✽ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❘❙❆ ✖ ❑❡② ❣❡♥❡r❛t✐♦♥

❑❡② ❣❡♥❡r❛t✐♦♥

✶ ❈❤♦♦s❡ t✇♦ ❞✐st✐♥❝t ❧❛r❣❡ ♣r✐♠❡ ♥✉♠❜❡rs p ✐ q✱ ✷ ❈❤♦♦s❡ ❛♥ ✐♥t❡❣❡r

❛♥❞ ✐s ❝♦♣r✐♠❡ ✇✐t❤ ✱

✸ ❈♦♠♣✉t❡

✱

✹ ❈♦♠♣✉t❡

✳ P✉❜❧✐❝ ❦❡②✿ ✳ Pr✐✈❛t❡ ❦❡②✿ ✳

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✾ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❘❙❆ ✖ ❑❡② ❣❡♥❡r❛t✐♦♥

❑❡② ❣❡♥❡r❛t✐♦♥

✶ ❈❤♦♦s❡ t✇♦ ❞✐st✐♥❝t ❧❛r❣❡ ♣r✐♠❡ ♥✉♠❜❡rs p ✐ q✱ ✷ ❈❤♦♦s❡ ❛♥ ✐♥t❡❣❡r e > 1 ❛♥❞ e ✐s ❝♦♣r✐♠❡ ✇✐t❤ (p − 1)(q − 1)✱ ✸ ❈♦♠♣✉t❡

✱

✹ ❈♦♠♣✉t❡

✳ P✉❜❧✐❝ ❦❡②✿ ✳ Pr✐✈❛t❡ ❦❡②✿ ✳

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✾ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❘❙❆ ✖ ❑❡② ❣❡♥❡r❛t✐♦♥

❑❡② ❣❡♥❡r❛t✐♦♥

✶ ❈❤♦♦s❡ t✇♦ ❞✐st✐♥❝t ❧❛r❣❡ ♣r✐♠❡ ♥✉♠❜❡rs p ✐ q✱ ✷ ❈❤♦♦s❡ ❛♥ ✐♥t❡❣❡r e > 1 ❛♥❞ e ✐s ❝♦♣r✐♠❡ ✇✐t❤ (p − 1)(q − 1)✱ ✸ ❈♦♠♣✉t❡ d = e−1mod(p − 1)(q − 1)✱ ✹ ❈♦♠♣✉t❡

✳ P✉❜❧✐❝ ❦❡②✿ ✳ Pr✐✈❛t❡ ❦❡②✿ ✳

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✾ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❘❙❆ ✖ ❑❡② ❣❡♥❡r❛t✐♦♥

❑❡② ❣❡♥❡r❛t✐♦♥

✶ ❈❤♦♦s❡ t✇♦ ❞✐st✐♥❝t ❧❛r❣❡ ♣r✐♠❡ ♥✉♠❜❡rs p ✐ q✱ ✷ ❈❤♦♦s❡ ❛♥ ✐♥t❡❣❡r e > 1 ❛♥❞ e ✐s ❝♦♣r✐♠❡ ✇✐t❤ (p − 1)(q − 1)✱ ✸ ❈♦♠♣✉t❡ d = e−1mod(p − 1)(q − 1)✱ ✹ ❈♦♠♣✉t❡ n = pq✳

P✉❜❧✐❝ ❦❡②✿ ✳ Pr✐✈❛t❡ ❦❡②✿ ✳

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✾ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❘❙❆ ✖ ❑❡② ❣❡♥❡r❛t✐♦♥

❑❡② ❣❡♥❡r❛t✐♦♥

✶ ❈❤♦♦s❡ t✇♦ ❞✐st✐♥❝t ❧❛r❣❡ ♣r✐♠❡ ♥✉♠❜❡rs p ✐ q✱ ✷ ❈❤♦♦s❡ ❛♥ ✐♥t❡❣❡r e > 1 ❛♥❞ e ✐s ❝♦♣r✐♠❡ ✇✐t❤ (p − 1)(q − 1)✱ ✸ ❈♦♠♣✉t❡ d = e−1mod(p − 1)(q − 1)✱ ✹ ❈♦♠♣✉t❡ n = pq✳

P✉❜❧✐❝ ❦❡②✿ (n, e)✳ Pr✐✈❛t❡ ❦❡②✿ ✳

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✾ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❘❙❆ ✖ ❑❡② ❣❡♥❡r❛t✐♦♥

❑❡② ❣❡♥❡r❛t✐♦♥

✶ ❈❤♦♦s❡ t✇♦ ❞✐st✐♥❝t ❧❛r❣❡ ♣r✐♠❡ ♥✉♠❜❡rs p ✐ q✱ ✷ ❈❤♦♦s❡ ❛♥ ✐♥t❡❣❡r e > 1 ❛♥❞ e ✐s ❝♦♣r✐♠❡ ✇✐t❤ (p − 1)(q − 1)✱ ✸ ❈♦♠♣✉t❡ d = e−1mod(p − 1)(q − 1)✱ ✹ ❈♦♠♣✉t❡ n = pq✳

P✉❜❧✐❝ ❦❡②✿ (n, e)✳ Pr✐✈❛t❡ ❦❡②✿ (n, d)✳

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✾ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❊♥❝r②♣t✐♦♥ ✇✐t❤ ❘❙❆

✶ ▼❡ss❛❣❡ ✐♥ ❞✐✈✐❞❡❞ ✐♥t♦ ❜❧♦❝❦s ♦❢ s✐③❡ ♥♦t ❣r❡❛t❡r t❤❛t ❦❡② s✐③❡ ✭✐✳❡✳

✷✵✹✽ ❜✐ts✮✳ ▼❡ss❛❣❡ ✐s tr❡❛t❡❞ ❛s ✐♥t❡❣❡r ♥✉♠❜❡r mi < n✳

✷ ❊♥❝r②♣t✐♦♥✿

ci = me

i mod n

✸ ❉❡❝r②♣t✐♦♥✿

mi = cd

i mod n✳

✹ ❘❙❆ ❡♥❝r②♣t✐♦♥ ✐♥ t❤✐s ❢♦r♠ ✐s ✐♥s❡❝✉r❡ ✖ t❡①t❜♦♦❦ ❘❙❆ P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✶✵ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❉✐❣✐t❛❧ s✐❣♥❛t✉r❡

❉✐❣✐t❛❧ s✐❣♥❛t✉r❡ ❝❛♥ ❜❡ ✐♠♣❧❡♠❡♥t❡❞ ✉s✐♥❣ ❘❙❆ ❝r②♣t♦s②st❡♠ ❆✉t❤♦r✿

✶ ❆♣♣❧✐❡s ❞✐❣❡st ❢✉♥❝t✐♦♥ t♦ t❤❡ ♠❡ss❛❣❡✿

✳

✷ ❙✐❣♥❛t✉r❡ ✐s ❣❡♥❡r❛t❡❞ ❜② ❡♥❝r②♣t✐♥❣ ❞✐❣❡st ✇✐t❤ ♣r✐✈❛t❡ ❦❡②✿ ✸ ❙❡♥❞ ♠❡ss❛❣❡ ❛♥❞ s✐❣♥❛t✉r❡ t♦ r❡❝❡✐✈❡r

❘❡❝❡✐✈❡r✿

✶ ❊♥❝r②♣ts s✐❣♥❛t✉r❡ ✇✐t❤ ♣✉❜❧✐❝ ❦❡②✿ ✷ ❈♦♠♣✉t❡ ❞✐❣❡st ❢♦r r❡❝❡✐✈❡❞ ♠❡ss❛❣❡✳ ✸ ❈♦♠♣❛r❡

❛♥❞ ✈❡r✐❢② t❤❡ s✐❣♥❛t✉r❡✳

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✶✶ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❉✐❣✐t❛❧ s✐❣♥❛t✉r❡

❉✐❣✐t❛❧ s✐❣♥❛t✉r❡ ❝❛♥ ❜❡ ✐♠♣❧❡♠❡♥t❡❞ ✉s✐♥❣ ❘❙❆ ❝r②♣t♦s②st❡♠ ❆✉t❤♦r✿

✶ ❆♣♣❧✐❡s ❞✐❣❡st ❢✉♥❝t✐♦♥ t♦ t❤❡ ♠❡ss❛❣❡✿ hm = hash(m)✳ ✷ ❙✐❣♥❛t✉r❡ ✐s ❣❡♥❡r❛t❡❞ ❜② ❡♥❝r②♣t✐♥❣ ❞✐❣❡st ✇✐t❤ ♣r✐✈❛t❡ ❦❡②✿ ✸ ❙❡♥❞ ♠❡ss❛❣❡ ❛♥❞ s✐❣♥❛t✉r❡ t♦ r❡❝❡✐✈❡r

❘❡❝❡✐✈❡r✿

✶ ❊♥❝r②♣ts s✐❣♥❛t✉r❡ ✇✐t❤ ♣✉❜❧✐❝ ❦❡②✿ ✷ ❈♦♠♣✉t❡ ❞✐❣❡st ❢♦r r❡❝❡✐✈❡❞ ♠❡ss❛❣❡✳ ✸ ❈♦♠♣❛r❡

❛♥❞ ✈❡r✐❢② t❤❡ s✐❣♥❛t✉r❡✳

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✶✶ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❉✐❣✐t❛❧ s✐❣♥❛t✉r❡

❉✐❣✐t❛❧ s✐❣♥❛t✉r❡ ❝❛♥ ❜❡ ✐♠♣❧❡♠❡♥t❡❞ ✉s✐♥❣ ❘❙❆ ❝r②♣t♦s②st❡♠ ❆✉t❤♦r✿

✶ ❆♣♣❧✐❡s ❞✐❣❡st ❢✉♥❝t✐♦♥ t♦ t❤❡ ♠❡ss❛❣❡✿ hm = hash(m)✳ ✷ ❙✐❣♥❛t✉r❡ ✐s ❣❡♥❡r❛t❡❞ ❜② ❡♥❝r②♣t✐♥❣ ❞✐❣❡st ✇✐t❤ ♣r✐✈❛t❡ ❦❡②✿

sigm = hd

mmod n

✸ ❙❡♥❞ ♠❡ss❛❣❡ ❛♥❞ s✐❣♥❛t✉r❡ t♦ r❡❝❡✐✈❡r

❘❡❝❡✐✈❡r✿

✶ ❊♥❝r②♣ts s✐❣♥❛t✉r❡ ✇✐t❤ ♣✉❜❧✐❝ ❦❡②✿ ✷ ❈♦♠♣✉t❡ ❞✐❣❡st ❢♦r r❡❝❡✐✈❡❞ ♠❡ss❛❣❡✳ ✸ ❈♦♠♣❛r❡

❛♥❞ ✈❡r✐❢② t❤❡ s✐❣♥❛t✉r❡✳

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✶✶ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❉✐❣✐t❛❧ s✐❣♥❛t✉r❡

❉✐❣✐t❛❧ s✐❣♥❛t✉r❡ ❝❛♥ ❜❡ ✐♠♣❧❡♠❡♥t❡❞ ✉s✐♥❣ ❘❙❆ ❝r②♣t♦s②st❡♠ ❆✉t❤♦r✿

✶ ❆♣♣❧✐❡s ❞✐❣❡st ❢✉♥❝t✐♦♥ t♦ t❤❡ ♠❡ss❛❣❡✿ hm = hash(m)✳ ✷ ❙✐❣♥❛t✉r❡ ✐s ❣❡♥❡r❛t❡❞ ❜② ❡♥❝r②♣t✐♥❣ ❞✐❣❡st ✇✐t❤ ♣r✐✈❛t❡ ❦❡②✿

sigm = hd

mmod n

✸ ❙❡♥❞ ♠❡ss❛❣❡ ❛♥❞ s✐❣♥❛t✉r❡ t♦ r❡❝❡✐✈❡r

❘❡❝❡✐✈❡r✿

✶ ❊♥❝r②♣ts s✐❣♥❛t✉r❡ ✇✐t❤ ♣✉❜❧✐❝ ❦❡②✿ ✷ ❈♦♠♣✉t❡ ❞✐❣❡st ❢♦r r❡❝❡✐✈❡❞ ♠❡ss❛❣❡✳ ✸ ❈♦♠♣❛r❡

❛♥❞ ✈❡r✐❢② t❤❡ s✐❣♥❛t✉r❡✳

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✶✶ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❉✐❣✐t❛❧ s✐❣♥❛t✉r❡

❉✐❣✐t❛❧ s✐❣♥❛t✉r❡ ❝❛♥ ❜❡ ✐♠♣❧❡♠❡♥t❡❞ ✉s✐♥❣ ❘❙❆ ❝r②♣t♦s②st❡♠ ❆✉t❤♦r✿

✶ ❆♣♣❧✐❡s ❞✐❣❡st ❢✉♥❝t✐♦♥ t♦ t❤❡ ♠❡ss❛❣❡✿ hm = hash(m)✳ ✷ ❙✐❣♥❛t✉r❡ ✐s ❣❡♥❡r❛t❡❞ ❜② ❡♥❝r②♣t✐♥❣ ❞✐❣❡st ✇✐t❤ ♣r✐✈❛t❡ ❦❡②✿

sigm = hd

mmod n

✸ ❙❡♥❞ ♠❡ss❛❣❡ ❛♥❞ s✐❣♥❛t✉r❡ t♦ r❡❝❡✐✈❡r

❘❡❝❡✐✈❡r✿

✶ ❊♥❝r②♣ts s✐❣♥❛t✉r❡ ✇✐t❤ ♣✉❜❧✐❝ ❦❡②✿ ✷ ❈♦♠♣✉t❡ ❞✐❣❡st ❢♦r r❡❝❡✐✈❡❞ ♠❡ss❛❣❡✳ ✸ ❈♦♠♣❛r❡

❛♥❞ ✈❡r✐❢② t❤❡ s✐❣♥❛t✉r❡✳

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✶✶ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❉✐❣✐t❛❧ s✐❣♥❛t✉r❡

❉✐❣✐t❛❧ s✐❣♥❛t✉r❡ ❝❛♥ ❜❡ ✐♠♣❧❡♠❡♥t❡❞ ✉s✐♥❣ ❘❙❆ ❝r②♣t♦s②st❡♠ ❆✉t❤♦r✿

✶ ❆♣♣❧✐❡s ❞✐❣❡st ❢✉♥❝t✐♦♥ t♦ t❤❡ ♠❡ss❛❣❡✿ hm = hash(m)✳ ✷ ❙✐❣♥❛t✉r❡ ✐s ❣❡♥❡r❛t❡❞ ❜② ❡♥❝r②♣t✐♥❣ ❞✐❣❡st ✇✐t❤ ♣r✐✈❛t❡ ❦❡②✿

sigm = hd

mmod n

✸ ❙❡♥❞ ♠❡ss❛❣❡ ❛♥❞ s✐❣♥❛t✉r❡ t♦ r❡❝❡✐✈❡r

❘❡❝❡✐✈❡r✿

✶ ❊♥❝r②♣ts s✐❣♥❛t✉r❡ ✇✐t❤ ♣✉❜❧✐❝ ❦❡②✿ sige

m = hed m

✷ ❈♦♠♣✉t❡ ❞✐❣❡st ❢♦r r❡❝❡✐✈❡❞ ♠❡ss❛❣❡✳ ✸ ❈♦♠♣❛r❡

❛♥❞ ✈❡r✐❢② t❤❡ s✐❣♥❛t✉r❡✳

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✶✶ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❉✐❣✐t❛❧ s✐❣♥❛t✉r❡

❉✐❣✐t❛❧ s✐❣♥❛t✉r❡ ❝❛♥ ❜❡ ✐♠♣❧❡♠❡♥t❡❞ ✉s✐♥❣ ❘❙❆ ❝r②♣t♦s②st❡♠ ❆✉t❤♦r✿

✶ ❆♣♣❧✐❡s ❞✐❣❡st ❢✉♥❝t✐♦♥ t♦ t❤❡ ♠❡ss❛❣❡✿ hm = hash(m)✳ ✷ ❙✐❣♥❛t✉r❡ ✐s ❣❡♥❡r❛t❡❞ ❜② ❡♥❝r②♣t✐♥❣ ❞✐❣❡st ✇✐t❤ ♣r✐✈❛t❡ ❦❡②✿

sigm = hd

mmod n

✸ ❙❡♥❞ ♠❡ss❛❣❡ ❛♥❞ s✐❣♥❛t✉r❡ t♦ r❡❝❡✐✈❡r

❘❡❝❡✐✈❡r✿

✶ ❊♥❝r②♣ts s✐❣♥❛t✉r❡ ✇✐t❤ ♣✉❜❧✐❝ ❦❡②✿ sige

m = hed m

✷ ❈♦♠♣✉t❡ ❞✐❣❡st ❢♦r r❡❝❡✐✈❡❞ ♠❡ss❛❣❡✳ ✸ ❈♦♠♣❛r❡

❛♥❞ ✈❡r✐❢② t❤❡ s✐❣♥❛t✉r❡✳

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✶✶ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❉✐❣✐t❛❧ s✐❣♥❛t✉r❡

❉✐❣✐t❛❧ s✐❣♥❛t✉r❡ ❝❛♥ ❜❡ ✐♠♣❧❡♠❡♥t❡❞ ✉s✐♥❣ ❘❙❆ ❝r②♣t♦s②st❡♠ ❆✉t❤♦r✿

✶ ❆♣♣❧✐❡s ❞✐❣❡st ❢✉♥❝t✐♦♥ t♦ t❤❡ ♠❡ss❛❣❡✿ hm = hash(m)✳ ✷ ❙✐❣♥❛t✉r❡ ✐s ❣❡♥❡r❛t❡❞ ❜② ❡♥❝r②♣t✐♥❣ ❞✐❣❡st ✇✐t❤ ♣r✐✈❛t❡ ❦❡②✿

sigm = hd

mmod n

✸ ❙❡♥❞ ♠❡ss❛❣❡ ❛♥❞ s✐❣♥❛t✉r❡ t♦ r❡❝❡✐✈❡r

❘❡❝❡✐✈❡r✿

✶ ❊♥❝r②♣ts s✐❣♥❛t✉r❡ ✇✐t❤ ♣✉❜❧✐❝ ❦❡②✿ sige

m = hed m

✷ ❈♦♠♣✉t❡ ❞✐❣❡st ❢♦r r❡❝❡✐✈❡❞ ♠❡ss❛❣❡✳ ✸ ❈♦♠♣❛r❡ hed

m = hm ❛♥❞ ✈❡r✐❢② t❤❡ s✐❣♥❛t✉r❡✳

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✶✶ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❙❡❝✉r✐t② ♦❢ ❘❙❆

✶ ❈r②♣t❛♥❛❧②st ✇❛♥ts t♦ ❦♥♦✇ t❤❡ ♣r✐✈❛t❡ ❡①♣♦♥❡♥t d✳ ✷ ❊❛s② t♦ ❝♦♠♣✉t❡

✇❤❡♥ ✐ ❛r❡ ❦♥♦✇♥✳

✸ ❋❛❝t♦r✐③❛t✐♦♥ ♦❢

✐s ❝♦♠♣✉t❛t✐♦♥❛❧❧② ♥♦t ❢❡❛s✐❜❧❡✳

✹ ❙♦♠❡ ♠❡t❤♦❞ ❛r❡ ✉s❛❜❧❡ ✇❤❡♥ ❦❡② ❝♦♠♣♦♥❡♥ts ❢✉❧✜❧❧ s♦♠❡

❝♦♥❞✐t✐♦♥s✳

✺ ◗✉❛♥t✉♠ ❝♦♠♣✉t✐♥❣ ❛♥❞ ❙❤♦r ❛❧❣♦r✐t❤♠✳ P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✶✷ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❙❡❝✉r✐t② ♦❢ ❘❙❆

✶ ❈r②♣t❛♥❛❧②st ✇❛♥ts t♦ ❦♥♦✇ t❤❡ ♣r✐✈❛t❡ ❡①♣♦♥❡♥t d✳ ✷ ❊❛s② t♦ ❝♦♠♣✉t❡ d ✇❤❡♥ p ✐ q ❛r❡ ❦♥♦✇♥✳ ✸ ❋❛❝t♦r✐③❛t✐♦♥ ♦❢

✐s ❝♦♠♣✉t❛t✐♦♥❛❧❧② ♥♦t ❢❡❛s✐❜❧❡✳

✹ ❙♦♠❡ ♠❡t❤♦❞ ❛r❡ ✉s❛❜❧❡ ✇❤❡♥ ❦❡② ❝♦♠♣♦♥❡♥ts ❢✉❧✜❧❧ s♦♠❡

❝♦♥❞✐t✐♦♥s✳

✺ ◗✉❛♥t✉♠ ❝♦♠♣✉t✐♥❣ ❛♥❞ ❙❤♦r ❛❧❣♦r✐t❤♠✳ P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✶✷ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❙❡❝✉r✐t② ♦❢ ❘❙❆

✶ ❈r②♣t❛♥❛❧②st ✇❛♥ts t♦ ❦♥♦✇ t❤❡ ♣r✐✈❛t❡ ❡①♣♦♥❡♥t d✳ ✷ ❊❛s② t♦ ❝♦♠♣✉t❡ d ✇❤❡♥ p ✐ q ❛r❡ ❦♥♦✇♥✳ ✸ ❋❛❝t♦r✐③❛t✐♦♥ ♦❢ n ✐s ❝♦♠♣✉t❛t✐♦♥❛❧❧② ♥♦t ❢❡❛s✐❜❧❡✳ ✹ ❙♦♠❡ ♠❡t❤♦❞ ❛r❡ ✉s❛❜❧❡ ✇❤❡♥ ❦❡② ❝♦♠♣♦♥❡♥ts ❢✉❧✜❧❧ s♦♠❡

❝♦♥❞✐t✐♦♥s✳

✺ ◗✉❛♥t✉♠ ❝♦♠♣✉t✐♥❣ ❛♥❞ ❙❤♦r ❛❧❣♦r✐t❤♠✳ P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✶✷ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❙❡❝✉r✐t② ♦❢ ❘❙❆

✶ ❈r②♣t❛♥❛❧②st ✇❛♥ts t♦ ❦♥♦✇ t❤❡ ♣r✐✈❛t❡ ❡①♣♦♥❡♥t d✳ ✷ ❊❛s② t♦ ❝♦♠♣✉t❡ d ✇❤❡♥ p ✐ q ❛r❡ ❦♥♦✇♥✳ ✸ ❋❛❝t♦r✐③❛t✐♦♥ ♦❢ n ✐s ❝♦♠♣✉t❛t✐♦♥❛❧❧② ♥♦t ❢❡❛s✐❜❧❡✳ ✹ ❙♦♠❡ ♠❡t❤♦❞ ❛r❡ ✉s❛❜❧❡ ✇❤❡♥ ❦❡② ❝♦♠♣♦♥❡♥ts ❢✉❧✜❧❧ s♦♠❡

❝♦♥❞✐t✐♦♥s✳

✺ ◗✉❛♥t✉♠ ❝♦♠♣✉t✐♥❣ ❛♥❞ ❙❤♦r ❛❧❣♦r✐t❤♠✳ P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✶✷ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❙❡❝✉r✐t② ♦❢ ❘❙❆

✶ ❈r②♣t❛♥❛❧②st ✇❛♥ts t♦ ❦♥♦✇ t❤❡ ♣r✐✈❛t❡ ❡①♣♦♥❡♥t d✳ ✷ ❊❛s② t♦ ❝♦♠♣✉t❡ d ✇❤❡♥ p ✐ q ❛r❡ ❦♥♦✇♥✳ ✸ ❋❛❝t♦r✐③❛t✐♦♥ ♦❢ n ✐s ❝♦♠♣✉t❛t✐♦♥❛❧❧② ♥♦t ❢❡❛s✐❜❧❡✳ ✹ ❙♦♠❡ ♠❡t❤♦❞ ❛r❡ ✉s❛❜❧❡ ✇❤❡♥ ❦❡② ❝♦♠♣♦♥❡♥ts ❢✉❧✜❧❧ s♦♠❡

❝♦♥❞✐t✐♦♥s✳

✺ ◗✉❛♥t✉♠ ❝♦♠♣✉t✐♥❣ ❛♥❞ ❙❤♦r ❛❧❣♦r✐t❤♠✳ P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✶✷ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❲❤② t❤❡ ❘❙❆ ✐s s♦ ✐♠♣♦rt❛♥t❄

❙②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✐s s❡❝✉r❡ ❜✉t ❤♦✇ t♦ ❡①❝❤❛♥❣❡ ❦❡②s❄ P✉❜❧✐❝✲❦❡② ❝r②♣t♦❣r❛♣❤② ✐s ❡①tr❡♠❡❧② s❧♦✇✳ ❙♦❧✉t✐♦♥✿

✶ ♣✉❜❧✐❝✲❦❡② ❝r②♣t♦❣r❛♣❤② ❢♦r ❦❡② ❡①❝❤❛♥❣❡✴♥❡❣♦t✐❛t✐♦♥ ✷ s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ❢♦r ❞❛t❛ ❡♥❝r②♣t✐♦♥✴❞❡❝r②♣t✐♦♥✳

❍♦✇ t♦ ❡♥s✉r❡ t❤❛t ♣✉❜❧✐❝ ❦❡② ✐s ❝♦rr❡❝t❄

✶ ❲❡❜ ♦❢ ❚r✉st ❧✐❦❡ ✐♥ P●P ✷ P✉❜❧✐❝ ❑❡② ■♥❢r❛str✉❝t✉r❡ P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✶✸ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❙❙▲✴❚▲❙ ♣r♦t♦❝♦❧

❙❡❝✉r❡❞ ❙♦❝❦❡t ▲❛②❡r ◆❡ts❝❛♣❡ ✶✾✾✹ ✭❙❙▲ ✸✳✵✱ ✶✾✾✺✮ ❚▲❙ ✶✳✵ ❚r❛♥s♣♦rt ▲❛②❡r ❙❡❝✉r✐t② ✶✾✾✻ ❚▲❙ ✶✳✷✱ ❘❋❈ ✺✷✹✻✱ ❙✐❡r♣✐❡➠ ✷✵✵✽✳ ❚▲❙ ✶✳✸✱ ✷✵✶✽✳ ❖♣❡♥❙❙▲✱ ●♥✉❚▲❙ ✖ ♦♣❡♥ ✐♠♣❧❡♠❡♥t❛t✐♦♥s

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✶✹ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❙❙▲✴❚▲❙ ♣r♦t♦❝♦❧

❙❡❝✉r❡❞ ❙♦❝❦❡t ▲❛②❡r ◆❡ts❝❛♣❡ ✶✾✾✹ ✭❙❙▲ ✸✳✵✱ ✶✾✾✺✮ ❚▲❙ ✶✳✵ ❚r❛♥s♣♦rt ▲❛②❡r ❙❡❝✉r✐t② ✶✾✾✻ ❚▲❙ ✶✳✷✱ ❘❋❈ ✺✷✹✻✱ ❙✐❡r♣✐❡➠ ✷✵✵✽✳ ❚▲❙ ✶✳✸✱ ✷✵✶✽✳ ❖♣❡♥❙❙▲✱ ●♥✉❚▲❙ ✖ ♦♣❡♥ ✐♠♣❧❡♠❡♥t❛t✐♦♥s

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✶✹ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❙❙▲✴❚▲❙ ♣r♦t♦❝♦❧

❙❡❝✉r❡❞ ❙♦❝❦❡t ▲❛②❡r ◆❡ts❝❛♣❡ ✶✾✾✹ ✭❙❙▲ ✸✳✵✱ ✶✾✾✺✮ ❚▲❙ ✶✳✵ ❚r❛♥s♣♦rt ▲❛②❡r ❙❡❝✉r✐t② ✶✾✾✻ ❚▲❙ ✶✳✷✱ ❘❋❈ ✺✷✹✻✱ ❙✐❡r♣✐❡➠ ✷✵✵✽✳ ❚▲❙ ✶✳✸✱ ✷✵✶✽✳ ❖♣❡♥❙❙▲✱ ●♥✉❚▲❙ ✖ ♦♣❡♥ ✐♠♣❧❡♠❡♥t❛t✐♦♥s

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✶✹ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❙❙▲✴❚▲❙ ♣r♦t♦❝♦❧

❙❡❝✉r❡❞ ❙♦❝❦❡t ▲❛②❡r ◆❡ts❝❛♣❡ ✶✾✾✹ ✭❙❙▲ ✸✳✵✱ ✶✾✾✺✮ ❚▲❙ ✶✳✵ ❚r❛♥s♣♦rt ▲❛②❡r ❙❡❝✉r✐t② ✶✾✾✻ ❚▲❙ ✶✳✷✱ ❘❋❈ ✺✷✹✻✱ ❙✐❡r♣✐❡➠ ✷✵✵✽✳ ❚▲❙ ✶✳✸✱ ✷✵✶✽✳ ❖♣❡♥❙❙▲✱ ●♥✉❚▲❙ ✖ ♦♣❡♥ ✐♠♣❧❡♠❡♥t❛t✐♦♥s

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✶✹ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❙❙▲✴❚▲❙ ♣r♦t♦❝♦❧

❙❡❝✉r❡❞ ❙♦❝❦❡t ▲❛②❡r ◆❡ts❝❛♣❡ ✶✾✾✹ ✭❙❙▲ ✸✳✵✱ ✶✾✾✺✮ ❚▲❙ ✶✳✵ ❚r❛♥s♣♦rt ▲❛②❡r ❙❡❝✉r✐t② ✶✾✾✻ ❚▲❙ ✶✳✷✱ ❘❋❈ ✺✷✹✻✱ ❙✐❡r♣✐❡➠ ✷✵✵✽✳ ❚▲❙ ✶✳✸✱ ✷✵✶✽✳ ❖♣❡♥❙❙▲✱ ●♥✉❚▲❙ ✖ ♦♣❡♥ ✐♠♣❧❡♠❡♥t❛t✐♦♥s

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✶✹ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❙❙▲✴❚▲❙ ♣r♦t♦❝♦❧

❙❡❝✉r❡❞ ❙♦❝❦❡t ▲❛②❡r ◆❡ts❝❛♣❡ ✶✾✾✹ ✭❙❙▲ ✸✳✵✱ ✶✾✾✺✮ ❚▲❙ ✶✳✵ ❚r❛♥s♣♦rt ▲❛②❡r ❙❡❝✉r✐t② ✶✾✾✻ ❚▲❙ ✶✳✷✱ ❘❋❈ ✺✷✹✻✱ ❙✐❡r♣✐❡➠ ✷✵✵✽✳ ❚▲❙ ✶✳✸✱ ✷✵✶✽✳ ❖♣❡♥❙❙▲✱ ●♥✉❚▲❙ ✖ ♦♣❡♥ ✐♠♣❧❡♠❡♥t❛t✐♦♥s

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✶✹ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❍❛♥❞s❤❛❦❡ ♣r♦t♦❝♦❧ ✇ ❙❙▲✴❚▲❙

Client Hello Server Hello Server Certificate Server Key Exchange Client Certificate Request Server Hello Done Client Certificate Client Key Exchange Certificate Verify Change Cipher Spec Client Finished Message Change Cipher Spec Server Finished Message

Handshake protocol Application Data Record protocol

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✶✺ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❈❧✐❡♥t✴❙❡r✈❡r ❍❡❧❧♦ ♠❡ss❛❣❡s

❈❧✐❡♥t ❍❡❧❧♦ ❙❡r✈❡r ❍❡❧❧♦ ❍✐❣❤❡st s✉♣♣♦rt❡❞ ❚▲❙ ✈❡rs✐♦♥✳ ✸✷❇ str✐♥❣✿ ✹❇❬❝❧✐❡♥t✬s t✐♠❡❪ ✰ ✷✽❇❬r❛♥❞♦♠ ♥✉♠❜❡r❪ ✖ ✇✐❧❧ ❜❡ ✉s❡❞ t♦ ❣❡♥❡r❛t❡ s❡ss✐♦♥ ❦❡②✳ ❙❡ss✐♦♥ ■❉ ✖ ♦♥❧② ✐❢ s❡ss✐♦♥ ✐s r❡st❛rt❡❞✳ ❙✉♣♣♦rt❡❞ ❝r②♣t♦❣r❛♣❤✐❝ s✉✐t❡✱ ♥♣✳ ❚▲❙❴❊❈❉❍❊❴❘❙❆❴❲■❚❍ ❴❆❊❙❴✶✷✽❴●❈▼❴❙❍❆✷✺✻✳ ⇐✳ ⇐✳ ❙❡ss✐♦♥ ■❉✿

♥❡✇ ■❉ ♦r ■❉ ❡①✐st✐♥❣ s❡ss✐♦♥ ♦r ♥✉❧❧ ✖ ♥♦t s✉♣♣♦rt❡❞

❈r②♣t♦❣r❛♣❤✐❝ s✉✐t❡✳ ❙❡r✈❡r ❝❤♦♦s❡s t❤❡ str♦♥❣❡st✳

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✶✻ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❈✐♣❤❡r s✉✐t❡s

❚▲❙❴❊❈❉❍❊❴❘❙❆❴❲■❚❍❴❆❊❙❴✶✷✽❴●❈▼❴❙❍❆✷✺✻✳ ❊❈❉❍❊ ✖ ❊❧❧✐♣t✐❝ ❝✉r✈❡ ❉✐✣❡✕❍❡❧❧♠❛♥ ❢♦r s❡ss✐♦♥ ❦❡② ❘❙❆ ✖ ❢♦r ❝❡rt✐✜❝❛t❡s ❛♥❞ ❞✐❣✐t❛❧ s✐❣♥❛t✉r❡✳ ❆❊❙❴✶✷✽❴●❈▼ ✖ ❆❊❙ ❡♥❝r②♣t✐♦♥✱ ✶✷✽❜ ❜❧♦❝❦✱ ●❈▼ ✭●❛❧♦✐s✴❈♦✉♥t❡r ▼♦❞❡✮ ❝✐♣❤❡r ♠♦❞❡✳ ❙❍❆✷✺✻ ✖ ❞✐❣❡st ❢✉♥❝t✐♦♥✳

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✶✼ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

■♥t❡❣❡r ❢❛❝t♦r✐③❛t✐♦♥

■♥♣✉t ❞❛t❡✿ ✐♥t❡❣❡r N ✇✐t❤ d ❞✐❣✐ts✳ ❚❛s❦✿ ✜♥❞ ❛❧❧ ♣r✐♠❡ ❢❛❝t♦rs ♦❢ ✳ ❇r❡❛❦✐♥❣ ❘❙❆ r❡q✉✐r❡s ✜♥❞✐♥❣ ♦♥❧② t✇♦ ♣r✐♠❡ ❢❛❝t♦rs ❇❡st ❝♦♥✈❡♥t✐♦♥❛❧ ❛❧❣♦r✐t❤♠ ❤❛s s✉❜✲❡①♣♦♥❡♥t✐❛❧ ❝♦♠♣✉t❛t✐♦♥❛❧ ❝♦♠♣❧❡①✐t②✿ ●❡♥❡r❛❧ ◆✉♠❜❡r ❋✐❡❧❞ ❙✐❡✈❡ ✭●◆❋❙✮ ❙❤♦r✬s ❛❧❣♦r✐t❤♠ ❤❛s ♣♦❧②♥♦♠✐❛❧ ❝♦♠♣❧❡①✐t②

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✶✽ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

■♥t❡❣❡r ❢❛❝t♦r✐③❛t✐♦♥

■♥♣✉t ❞❛t❡✿ ✐♥t❡❣❡r N ✇✐t❤ d ❞✐❣✐ts✳ ❚❛s❦✿ ✜♥❞ ❛❧❧ ♣r✐♠❡ ❢❛❝t♦rs ♦❢ N✳ ❇r❡❛❦✐♥❣ ❘❙❆ r❡q✉✐r❡s ✜♥❞✐♥❣ ♦♥❧② t✇♦ ♣r✐♠❡ ❢❛❝t♦rs ❇❡st ❝♦♥✈❡♥t✐♦♥❛❧ ❛❧❣♦r✐t❤♠ ❤❛s s✉❜✲❡①♣♦♥❡♥t✐❛❧ ❝♦♠♣✉t❛t✐♦♥❛❧ ❝♦♠♣❧❡①✐t②✿ ●❡♥❡r❛❧ ◆✉♠❜❡r ❋✐❡❧❞ ❙✐❡✈❡ ✭●◆❋❙✮ ❙❤♦r✬s ❛❧❣♦r✐t❤♠ ❤❛s ♣♦❧②♥♦♠✐❛❧ ❝♦♠♣❧❡①✐t②

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✶✽ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

■♥t❡❣❡r ❢❛❝t♦r✐③❛t✐♦♥

■♥♣✉t ❞❛t❡✿ ✐♥t❡❣❡r N ✇✐t❤ d ❞✐❣✐ts✳ ❚❛s❦✿ ✜♥❞ ❛❧❧ ♣r✐♠❡ ❢❛❝t♦rs ♦❢ N✳ ❇r❡❛❦✐♥❣ ❘❙❆ r❡q✉✐r❡s ✜♥❞✐♥❣ ♦♥❧② t✇♦ ♣r✐♠❡ ❢❛❝t♦rs ❇❡st ❝♦♥✈❡♥t✐♦♥❛❧ ❛❧❣♦r✐t❤♠ ❤❛s s✉❜✲❡①♣♦♥❡♥t✐❛❧ ❝♦♠♣✉t❛t✐♦♥❛❧ ❝♦♠♣❧❡①✐t②✿ ●❡♥❡r❛❧ ◆✉♠❜❡r ❋✐❡❧❞ ❙✐❡✈❡ ✭●◆❋❙✮ ❙❤♦r✬s ❛❧❣♦r✐t❤♠ ❤❛s ♣♦❧②♥♦♠✐❛❧ ❝♦♠♣❧❡①✐t②

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✶✽ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

■♥t❡❣❡r ❢❛❝t♦r✐③❛t✐♦♥

■♥♣✉t ❞❛t❡✿ ✐♥t❡❣❡r N ✇✐t❤ d ❞✐❣✐ts✳ ❚❛s❦✿ ✜♥❞ ❛❧❧ ♣r✐♠❡ ❢❛❝t♦rs ♦❢ N✳ ❇r❡❛❦✐♥❣ ❘❙❆ r❡q✉✐r❡s ✜♥❞✐♥❣ ♦♥❧② t✇♦ ♣r✐♠❡ ❢❛❝t♦rs ❇❡st ❝♦♥✈❡♥t✐♦♥❛❧ ❛❧❣♦r✐t❤♠ ❤❛s s✉❜✲❡①♣♦♥❡♥t✐❛❧ ❝♦♠♣✉t❛t✐♦♥❛❧ ❝♦♠♣❧❡①✐t②✿ ●❡♥❡r❛❧ ◆✉♠❜❡r ❋✐❡❧❞ ❙✐❡✈❡ ✭●◆❋❙✮ ❙❤♦r✬s ❛❧❣♦r✐t❤♠ ❤❛s ♣♦❧②♥♦♠✐❛❧ ❝♦♠♣❧❡①✐t②

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✶✽ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

■♥t❡❣❡r ❢❛❝t♦r✐③❛t✐♦♥

■♥♣✉t ❞❛t❡✿ ✐♥t❡❣❡r N ✇✐t❤ d ❞✐❣✐ts✳ ❚❛s❦✿ ✜♥❞ ❛❧❧ ♣r✐♠❡ ❢❛❝t♦rs ♦❢ N✳ ❇r❡❛❦✐♥❣ ❘❙❆ r❡q✉✐r❡s ✜♥❞✐♥❣ ♦♥❧② t✇♦ ♣r✐♠❡ ❢❛❝t♦rs ❇❡st ❝♦♥✈❡♥t✐♦♥❛❧ ❛❧❣♦r✐t❤♠ ❤❛s s✉❜✲❡①♣♦♥❡♥t✐❛❧ ❝♦♠♣✉t❛t✐♦♥❛❧ ❝♦♠♣❧❡①✐t②✿ ●❡♥❡r❛❧ ◆✉♠❜❡r ❋✐❡❧❞ ❙✐❡✈❡ ✭●◆❋❙✮ ❙❤♦r✬s ❛❧❣♦r✐t❤♠ ❤❛s ♣♦❧②♥♦♠✐❛❧ ❝♦♠♣❧❡①✐t②

❙♦✉r❝❡✿ ■❇▼ ◗✉❛♥t✉♠ ❊①♣❡r✐❡♥❝❡ P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✶✽ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❙❤♦r ❛❧❣♦r✐t❤♠

P❡t❡r ❙❤♦r✱ ✶✾✾✹

✶ ❈❧❛ss✐❝ ✭❝♦♥✈❡♥t✐♦♥❛❧✮ ♣❛rt✿ ✐♥t❡❣❡r ❢❛❝t♦r✐③❛t✐♦♥ t♦ ♦r❞❡r✲✜♥❞✐♥❣

♣r♦❜❧❡♠

✷ ◗✉❛♥t✉♠ ♣❛rt✿ s♦❧✈✐♥❣ ♦r❞❡r✲✜♥❞✐♥❣ ♣r♦❜❧❡♠ P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✶✾ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

Pr♦❜❧❡♠ ♦❢ ♦r❞❡r✲✜♥❞✐♥❣

▼♦❞✉❧♦ ♦♣❡r❛t✐♦♥✿ a ≡ b (mod N) t✇♦ ✐♥t❡❣❡rs a✱ b ❛r❡ ❝♦♥❣r✉❡♥t ♠♦❞✉❧♦ N ✐❢ t❤❡r❡ ✐s ❛♥ ✐♥t❡❣❡r k s✉❝❤ t❤❛t a − b = kn ▲❡t✬s ZN = {0, . . . N − 1} ❜❡ ❛ s❡t ❞❡✜♥❡❞ ❜② ♦♣❡r❛t✐♦♥s ♠♦❞✉❧♦ N ▲❡t✬s Z∗

N = {a ∈ ZN : GCD(a, N) = 1} ❜❡ t❤❡ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ❣r♦✉♣

♦❢ ✐♥t❡❣❡rs ♠♦❞✉❧♦ N ✭❛♣♣r♦♣r✐❛t❡ ♦♣❡r❛t✐♦♥s ❛r❡ ❞❡✜♥❡❞✮ ❋♦r a ∈ Z∗

N t❤❡ ♦r❞❡r ♦❢ a ✐♥ Z∗ N ✐s t❤❡ s♠❛❧❧❡st ♣♦s✐t✐✈❡ ✐♥t❡❣❡r r

s✉❝❤ t❤❛t✿ ar ≡ 1 (mod N)

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✷✵ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

Pr♦❜❧❡♠ ♦❢ ♦r❞❡r✲✜♥❞✐♥❣✿ ❡①❛♠♣❧❡

▲❡t✬s N = 15 ❛♥❞ a = 7✿ 7x mod N 71 (mod 15) = 7 72 (mod 15) = 4 73 (mod 15) = 13 74 (mod 15) = 1 ✭❊✉r❡❦❛✦✮ 75 (mod 15) = 16807 (mod 15) = 7 76 (mod 15) = 117649 (mod 15) = 4 77 (mod 15) = 823543 (mod 15) = 13 78 (mod 15) = 5764801(mod 15) = 1 ❖♥❝❡ ❛❣❛✐♥✦ t❤❡♥ ♦r❞❡r ♦❢ ✼ ✐♥ Z∗

15 ✐s r = 4✿

▲❡t✬s ❞❡♥♦t❡ t❤❡ ❛❜♦✈❡ ❡①♣r❡ss✐♦♥ ❛s ❢✉♥❝t✐♦♥ f7(x) = 7x (mod 15)✳ ■t ✐s ♣❡r✐♦❞✐❝❛❧✿ f7(x + 4) = f7(x)✳

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✷✶ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❋r♦♠ ✐♥t❡❣❡r ❢❛❝t♦r✐③❛t✐♦♥ t♦ ♦r❞❡r✲✜♥❞✐♥❣ ♣r♦❜❧❡♠

✶ ▲❡t✬s ❛ss✉♠❡ t❤❛t N ❤❛s t✇♦ ♣r✐♠❡ ❢❛❝t♦rs p1 ✐ p2✿ N = p1 × p2 ✷ P✐❝❦ ❛ r❛♥❞♦♠ ✐♥t❡❣❡r

✿ ✳

✸ ▲❡t✬s ❛ss✉♠❡ t❤❛t

✳

✹ ▲❡t✬s

✐s ♣❡r✐♦❞ ♦❢ ♠❛❣✐❝❛❧❧② ❝❛❧❝✉❧❛t❡❞

✺ ■❢

✐s ♦❞❞ ❣♦ ❜❛❝❦ t♦ st❡♣ ✷✳

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✷✷ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❋r♦♠ ✐♥t❡❣❡r ❢❛❝t♦r✐③❛t✐♦♥ t♦ ♦r❞❡r✲✜♥❞✐♥❣ ♣r♦❜❧❡♠

✶ ▲❡t✬s ❛ss✉♠❡ t❤❛t N ❤❛s t✇♦ ♣r✐♠❡ ❢❛❝t♦rs p1 ✐ p2✿ N = p1 × p2 ✷ P✐❝❦ ❛ r❛♥❞♦♠ ✐♥t❡❣❡r a✿ 2 ≥ a ≥ N − 1✳ ✸ ▲❡t✬s ❛ss✉♠❡ t❤❛t

✳

✹ ▲❡t✬s

✐s ♣❡r✐♦❞ ♦❢ ♠❛❣✐❝❛❧❧② ❝❛❧❝✉❧❛t❡❞

✺ ■❢

✐s ♦❞❞ ❣♦ ❜❛❝❦ t♦ st❡♣ ✷✳

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✷✷ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❋r♦♠ ✐♥t❡❣❡r ❢❛❝t♦r✐③❛t✐♦♥ t♦ ♦r❞❡r✲✜♥❞✐♥❣ ♣r♦❜❧❡♠

✶ ▲❡t✬s ❛ss✉♠❡ t❤❛t N ❤❛s t✇♦ ♣r✐♠❡ ❢❛❝t♦rs p1 ✐ p2✿ N = p1 × p2 ✷ P✐❝❦ ❛ r❛♥❞♦♠ ✐♥t❡❣❡r a✿ 2 ≥ a ≥ N − 1✳ ✸ ▲❡t✬s ❛ss✉♠❡ t❤❛t GCD(N, a) = 1✳ ✹ ▲❡t✬s

✐s ♣❡r✐♦❞ ♦❢ ♠❛❣✐❝❛❧❧② ❝❛❧❝✉❧❛t❡❞

✺ ■❢

✐s ♦❞❞ ❣♦ ❜❛❝❦ t♦ st❡♣ ✷✳

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✷✷ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❋r♦♠ ✐♥t❡❣❡r ❢❛❝t♦r✐③❛t✐♦♥ t♦ ♦r❞❡r✲✜♥❞✐♥❣ ♣r♦❜❧❡♠

✶ ▲❡t✬s ❛ss✉♠❡ t❤❛t N ❤❛s t✇♦ ♣r✐♠❡ ❢❛❝t♦rs p1 ✐ p2✿ N = p1 × p2 ✷ P✐❝❦ ❛ r❛♥❞♦♠ ✐♥t❡❣❡r a✿ 2 ≥ a ≥ N − 1✳ ✸ ▲❡t✬s ❛ss✉♠❡ t❤❛t GCD(N, a) = 1✳ ✹ ▲❡t✬s r ✐s ♣❡r✐♦❞ ♦❢ fa(x) = ax mod N ♠❛❣✐❝❛❧❧② ❝❛❧❝✉❧❛t❡❞ ✺ ■❢

✐s ♦❞❞ ❣♦ ❜❛❝❦ t♦ st❡♣ ✷✳

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✷✷ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❋r♦♠ ✐♥t❡❣❡r ❢❛❝t♦r✐③❛t✐♦♥ t♦ ♦r❞❡r✲✜♥❞✐♥❣ ♣r♦❜❧❡♠

✶ ▲❡t✬s ❛ss✉♠❡ t❤❛t N ❤❛s t✇♦ ♣r✐♠❡ ❢❛❝t♦rs p1 ✐ p2✿ N = p1 × p2 ✷ P✐❝❦ ❛ r❛♥❞♦♠ ✐♥t❡❣❡r a✿ 2 ≥ a ≥ N − 1✳ ✸ ▲❡t✬s ❛ss✉♠❡ t❤❛t GCD(N, a) = 1✳ ✹ ▲❡t✬s r ✐s ♣❡r✐♦❞ ♦❢ fa(x) = ax mod N ♠❛❣✐❝❛❧❧② ❝❛❧❝✉❧❛t❡❞ ✺ ■❢ r ✐s ♦❞❞ ❣♦ ❜❛❝❦ t♦ st❡♣ ✷✳ P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✷✷ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❋r♦♠ ✐♥t❡❣❡r ❢❛❝t♦r✐③❛t✐♦♥ ♦r❞❡r✲✜♥❞✐♥❣ ♣r♦❜❧❡♠ t

✶ ar ≡ 1 (mod N) ✷ ▲❡t✬s r❡❢♦r♠✉❧❛t❡✿ ✸ ❲❡ ♥♦t❡ t❤❛t

✐s ♥♦t ♠✉❧t✐♣❧② ♦❢ ✭ ✐s ♥♦t ❛ ♣❡r✐♦❞✮

✹ ▲❡t✬s ❛ss✉♠❡ t❤❛t

✐s ♥♦t ♠✉❧t✐♣❧② ♦❢ t♦♦✳

✺

❛r❡ ♥♦t ♠✉❧t✐♣❧② ♦❢ ❜✉t t❤❡② ♣r♦❞✉❝t ✐s✳

✻ Pr✐♠❡ ❢❛❝t♦rs✿ P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✷✸ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❋r♦♠ ✐♥t❡❣❡r ❢❛❝t♦r✐③❛t✐♦♥ ♦r❞❡r✲✜♥❞✐♥❣ ♣r♦❜❧❡♠ t

✶ ar ≡ 1 (mod N) ✷ ▲❡t✬s r❡❢♦r♠✉❧❛t❡✿ ar − 1 = (a r 2 − 1)(a r 2 + 1) ✸ ❲❡ ♥♦t❡ t❤❛t

✐s ♥♦t ♠✉❧t✐♣❧② ♦❢ ✭ ✐s ♥♦t ❛ ♣❡r✐♦❞✮

✹ ▲❡t✬s ❛ss✉♠❡ t❤❛t

✐s ♥♦t ♠✉❧t✐♣❧② ♦❢ t♦♦✳

✺

❛r❡ ♥♦t ♠✉❧t✐♣❧② ♦❢ ❜✉t t❤❡② ♣r♦❞✉❝t ✐s✳

✻ Pr✐♠❡ ❢❛❝t♦rs✿ P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✷✸ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❋r♦♠ ✐♥t❡❣❡r ❢❛❝t♦r✐③❛t✐♦♥ ♦r❞❡r✲✜♥❞✐♥❣ ♣r♦❜❧❡♠ t

✶ ar ≡ 1 (mod N) ✷ ▲❡t✬s r❡❢♦r♠✉❧❛t❡✿ ar − 1 = (a r 2 − 1)(a r 2 + 1) ✸ ❲❡ ♥♦t❡ t❤❛t a r 2 − 1 ✐s ♥♦t ♠✉❧t✐♣❧② ♦❢ N ✭ r

2 ✐s ♥♦t ❛ ♣❡r✐♦❞✮

✹ ▲❡t✬s ❛ss✉♠❡ t❤❛t

✐s ♥♦t ♠✉❧t✐♣❧② ♦❢ t♦♦✳

✺

❛r❡ ♥♦t ♠✉❧t✐♣❧② ♦❢ ❜✉t t❤❡② ♣r♦❞✉❝t ✐s✳

✻ Pr✐♠❡ ❢❛❝t♦rs✿ P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✷✸ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❋r♦♠ ✐♥t❡❣❡r ❢❛❝t♦r✐③❛t✐♦♥ ♦r❞❡r✲✜♥❞✐♥❣ ♣r♦❜❧❡♠ t

✶ ar ≡ 1 (mod N) ✷ ▲❡t✬s r❡❢♦r♠✉❧❛t❡✿ ar − 1 = (a r 2 − 1)(a r 2 + 1) ✸ ❲❡ ♥♦t❡ t❤❛t a r 2 − 1 ✐s ♥♦t ♠✉❧t✐♣❧② ♦❢ N ✭ r

2 ✐s ♥♦t ❛ ♣❡r✐♦❞✮

✹ ▲❡t✬s ❛ss✉♠❡ t❤❛t a r 2 + 1 ✐s ♥♦t ♠✉❧t✐♣❧② ♦❢ N t♦♦✳ ✺

❛r❡ ♥♦t ♠✉❧t✐♣❧② ♦❢ ❜✉t t❤❡② ♣r♦❞✉❝t ✐s✳

✻ Pr✐♠❡ ❢❛❝t♦rs✿ P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✷✸ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❋r♦♠ ✐♥t❡❣❡r ❢❛❝t♦r✐③❛t✐♦♥ ♦r❞❡r✲✜♥❞✐♥❣ ♣r♦❜❧❡♠ t

✶ ar ≡ 1 (mod N) ✷ ▲❡t✬s r❡❢♦r♠✉❧❛t❡✿ ar − 1 = (a r 2 − 1)(a r 2 + 1) ✸ ❲❡ ♥♦t❡ t❤❛t a r 2 − 1 ✐s ♥♦t ♠✉❧t✐♣❧② ♦❢ N ✭ r

2 ✐s ♥♦t ❛ ♣❡r✐♦❞✮

✹ ▲❡t✬s ❛ss✉♠❡ t❤❛t a r 2 + 1 ✐s ♥♦t ♠✉❧t✐♣❧② ♦❢ N t♦♦✳ ✺ a r 2 ± 1 ❛r❡ ♥♦t ♠✉❧t✐♣❧② ♦❢ N ❜✉t t❤❡② ♣r♦❞✉❝t ✐s✳ ✻ Pr✐♠❡ ❢❛❝t♦rs✿ P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✷✸ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❋r♦♠ ✐♥t❡❣❡r ❢❛❝t♦r✐③❛t✐♦♥ ♦r❞❡r✲✜♥❞✐♥❣ ♣r♦❜❧❡♠ t

✶ ar ≡ 1 (mod N) ✷ ▲❡t✬s r❡❢♦r♠✉❧❛t❡✿ ar − 1 = (a r 2 − 1)(a r 2 + 1) ✸ ❲❡ ♥♦t❡ t❤❛t a r 2 − 1 ✐s ♥♦t ♠✉❧t✐♣❧② ♦❢ N ✭ r

2 ✐s ♥♦t ❛ ♣❡r✐♦❞✮

✹ ▲❡t✬s ❛ss✉♠❡ t❤❛t a r 2 + 1 ✐s ♥♦t ♠✉❧t✐♣❧② ♦❢ N t♦♦✳ ✺ a r 2 ± 1 ❛r❡ ♥♦t ♠✉❧t✐♣❧② ♦❢ N ❜✉t t❤❡② ♣r♦❞✉❝t ✐s✳ ✻ Pr✐♠❡ ❢❛❝t♦rs✿ p1(p2) = GCD(N, a r 2 ± 1) P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✷✸ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❙❤♦r ❛❧❣♦r✐t❤♠

✶ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠✿ r❡tr✐❡✈❡ ❢r♦♠ ♣❡r✐♦❞✐❝ s✐❣♥❛❧ ❛❧❧ ❢r❡q✉❡♥❝✐❡s ✷ ◗✉❛♥t✉♠ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠ ✭◗❋❚✮ ❞♦❡s t❤❡ s❛♠❡ P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✷✹ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❙❤♦r✬s ❛❧❣♦r✐t❤♠ ✐♥ ♣r❛❝t✐❝❡

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✷✺ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❲❤❛t ❛❜♦✉t ♦t❤❡r ♣✉❜❧✐❝✲❦❡② ❝r②♣t♦s②st❡♠s❄

❉✐✣❡✲❍❡❧❧♠❛♥ ♣r♦t♦❝♦❧✱ ❊❧●❛♠❛❧ ❛♥❞ ❉❙❆✿ P❡t❡r ❲✳ ❙❤♦r✱ P♦❧②♥♦♠✐❛❧✲❚✐♠❡ ❆❧❣♦r✐t❤♠s ❢♦r Pr✐♠❡ ❋❛❝t♦r✐③❛t✐♦♥ ❛♥❞ ❉✐s❝r❡t❡ ▲♦❣❛r✐t❤♠s ♦♥ ❛ ◗✉❛♥t✉♠ ❈♦♠♣✉t❡r ❊❧❧✐♣t✐❝✲❈✉r✈❡ ❈r②♣t♦❣r❛♣❤② ✭❊❈❈✮✱ ❊❈❉❍ ✭❊❧❧✐♣t✐❝ ❈✉r✈❡ ❉✐✣❡✲❍❡❧❧♠❛♥✮✱ ❊❈❉❙❆ ✭❊❧❧✐♣t✐❝ ❈✉r✈❡ ❉✐❣✐t❛❧ ❙✐❣♥❛t✉r❡ ❆❧❣♦r✐t❤♠✮✿ ◗✉❛♥t✉♠ ❘❡s♦✉r❝❡ ❊st✐♠❛t❡s ❢♦r ❈♦♠♣✉t✐♥❣ ❊❧❧✐♣t✐❝ ❈✉r✈❡ ❉✐s❝r❡t❡ ▲♦❣❛r✐t❤♠s✱ ❙❤♦r✬s ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠ q✉❛♥t✉♠ ❛❧❣♦r✐t❤♠ ❢♦r ❡❧❧✐♣t✐❝ ❝✉r✈❡s

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✷✻ ✴ ✷✽

❘✐✈❡st✲❙❤❛♠✐r✲❆❞❡❧♠❛♥♥ ❈r②♣t♦s②st❡♠

❋✉t✉r❡ ♦❢ ❝r②♣t♦❣r❛♣❤②

❈r②♣t♦✲❛❣✐❧✐t② P♦st✲q✉❛♥t✉♠ ❝r②♣t♦❣r❛♣❤②✿

❧❛tt✐❝❡✲❜❛s❡❞ ❝r②♣t♦❣r❛♣❤② ❤❛s❤✲❜❛s❡❞ ❝r②♣t♦❣r❛♣❤② ❝♦rr❡❝t✐♦♥ ❝♦❞❡s✲❜❛s❡❞ ❝r②♣t♦❣r❛♣❤② ♠✉❧t✐✈❛r✐❛t❡ ❝r②♣t♦❣r❛♣❤②

P❛✇❡➟ ❚♦♣❛✱ P❤✳❉✳ ❑r❛❦♦✇ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐❝s ❙❡♠✐♥❛r ✭❑◗■❙✮ ❲❤❡♥ t❤❡ ❛s②♠♠❡tr✐❝ ❝r②♣t♦❣r❛♣❤② ✇✐❧❧ ❜❡ ♦✉t❞❛t❡❞❄ ❏✉♥❡ ✶✻✱ ✷✵✷✵ ✷✼ ✴ ✷✽